If and are the roots of the equation 3 x^2+2 x+1=0 , then the equation whose roots are + ⁻¹ and + ⁻¹ is

Step-by-step solution

3 x^2+2 x+1=0 Sum of the roots = + = -2 3 (1) Product of the roots = = 1 3 (2) We have to find the equation with the roots + ⁻¹ and + ⁻¹ . Sum of the roots (S)= + ⁻¹+ + ⁻¹= + 1 + + 1 aligned & =( + )+ ( + ) & = -2 3 + ( -2 3 1 3 ) ( from (1),(2)) aligned = -2 3 -2= -2-6 3 = -8 3 . Product of the roots ( P )= ( + ⁻¹ ) ( + ⁻¹ )= ( + 1 ) ( + 1 )= +1+1+ 1 = +2+ 1 = 1 3 +2+ 1 1 3 = 1 3 +2+3= 1 3 +5= 16 3 . So, the required equation is x^2-S . x .+P=0=x^2+ 8 3 x+ 16 3 =0 3 x^2+8 x+16=0